A Tree Model for Pricing Convertible Bonds with Equity, Interest Rate, and Default Risk

Abstract

Black and Scholes modeled stock options assuming the underlying stock was exposed to a single source of risk. Once it is recognized that both stocks and bonds are actually derivatives written on the underlying firm, many kinds of hybrid securities, like convertible bonds, are seen to have exposure to both stock market risk and interest rate risk. Further, as attention has turned to default as a critical risk, it is now apparent that properly valuing and risk-managing these instruments requires dealing formally with that third risk factor, as well. Developing tractable models for these cases seems like a daunting task, but Chambers and Lu present a lattice-based approach that does it. Their approach is easily implemented; it embeds important real world features of the problem, including correlation between stock price and interest rate changes; and it shares the advantage of other tree-based numerical methods, that increasing the number of time steps to improve accuracy does not cause execution time to explode in an unmanageable way. A real world example illustrates the use of the technique to price a convertible bond issued by Lucent.